1. Technical Field
The present invention relates to the field of fixed-point digital signal processing. Applications may be found for it in any fixed-point digital system, and particularly in the digitally modulated synthesizers used in the radio transmitters and the radio transceivers of a digital radiocommunication system.
2. Related Art
For carrying out operations on binary numbers, a floating-point digital system comprises software resources such as a correctly programmed DSP (Digital Signal Processor). A fixed-point system, however, only comprises sequential logic circuits such as digital adders, digital multipliers, shift registers or the like.
The binary numbers which are processed by a fixed-point digital system encode quantized values corresponding to a real value X (for example the variable value of the radio signal received by a radio receiver, or the constant value of the frequency of a radio channel). These quantized numbers are represented by integers between zero and 2n−1—where n is the number of bits used for encoding the information—if the value X is always positive, or between −(2n−1) and 2n−1 if the value X is signed (that is, if it can be negative). By convention, Xq denotes the quantized value which is obtained from the real value X by a quantizing operation. For linear quantizing, the correspondence between the real value X (so-called real information) and the quantized value Xq (so-called quantized information) is given by the relation:Xq=round(X×Cq)   (1)
where Cq is a real number referred to as the quantization coefficient.
The quantization of the system is determined by the number Cq in relation with the number n. The quantization coefficient Cq is such that:
                    {                                                                                                  round                    ⁢                                                                                  ⁢                                          (                                              |                                                  X                          ⁡                                                      (                            t                            )                                                                          |                                                  ×                          Cq                                                                    )                                                        ≤                                                            2                                              n                        -                        1                                                              -                    1                                                  ,                                  ∀                  t                                ,                                  if                  ⁢                                                                          ⁢                  the                  ⁢                                                                          ⁢                  information                  ⁢                                                                          ⁢                  X                  ⁢                                                                          ⁢                  is                  ⁢                                                                          ⁢                  signed                                                                                                                                              round                    ⁢                                                                                  ⁢                                          (                                                                        X                          ⁡                                                      (                            t                            )                                                                          ×                        Cq                                            )                                                        ≤                                                            2                      n                                        -                    1                                                  ,                                  ∀                  t                                ,                otherwise                                                                        (        2        )            where |x| denotes the absolute value operator of the real variable x.
The act of quantizing the information X creates an error, referred to as the quantization error and denoted by e, such that:
                    e        =                              X            -                          Xq              Cq                                =                      X            -                                          round                ⁡                                  (                                      X                    ×                    Cq                                    )                                            Cq                                                          (        3        )            
The error e is of course variable, inasmuch as it depends on the value X. According to the properties of the rounding function, the error e is always such that
          e        ≤            1              2        ×        Cq              .  The maximum value of the quantization error, denoted by emax, is therefore given by:
                              e          max                =                  1                      2            ×            Cq                                              (        4        )            
The inverse of the quantization coefficient Cq is the resolution of the digital system, that is to say the smallest variation of the real information which is distinguishable in the quantized information. Put another way,
  1  Cqis such that if
  X  =            1      Cq        +          X      ′      then Xq=1+Xq′.
Optimization of the dynamic range of the system generally leads to the quantization being defined by choosing Cq such that:
                    {                                                                              Cq                  =                                                            max                      ⁡                                              (                                                  |                                                      X                            ⁡                                                          (                              t                              )                                                                                |                                                )                                                                                                            2                                                  n                          -                          1                                                                    -                      1                                                                      ,                                  ∀                  t                                ,                                  if                  ⁢                                                                          ⁢                  the                  ⁢                                                                          ⁢                  information                  ⁢                                                                          ⁢                  X                  ⁢                                                                          ⁢                  is                  ⁢                                                                          ⁢                  signed                                                                                                                          Cq                  =                                                            max                      ⁡                                              (                                                  X                          ⁡                                                      (                            t                            )                                                                          )                                                                                                            2                        n                                            -                      1                                                                      ,                                  ∀                  t                                ,                otherwise                                                                        (        5        )            
Certain systems dictate the quantization of the digital data, for example in order to be compatible with analog signals after digital-analog conversion of a quantized signal. In this case, there is a quantization error majored in modulus by
            e      max        =          1              2        ×        Cq              ,where Cq is the corresponding quantization coefficient. However, it may arise that this resolution is insufficient for representing some or all of the digital signals of the system.
On the other hand, certain digital systems use constant digital values. In a radio transmitter or receiver, for example, such a digital constant may represent the central frequency of a radio channel. In this case, the situation may be encountered in which a quantization error affecting the digital constant (this error being systematic inasmuch as it does not vary) exceeds the maximum tolerable error for digital representation of this constant. If the system does not dictate the quantization of the digital data, then the systematic quantization error affecting a specific digital constant K may be reduced, albeit this may mean that the dynamic range of the system is not optimized, by choosing the quantization coefficient Cq such that
            K      -                        round          ⁡                      (                          K              ×              Cq                        )                          Cq              ≤          e      d        ≤          e      max        ,where ed is the maximum tolerable error for digital representation of the constant K. This is not always possible in a system which dictates the quantization of the digital data, such as a digitally modulated frequency synthesizer, for example.